Problem: Simplify the following expression: $ a = \dfrac{4}{9} - \dfrac{9r - 2}{-4r + 7} $
Explanation: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-4r + 7}{-4r + 7}$ $ \dfrac{4}{9} \times \dfrac{-4r + 7}{-4r + 7} = \dfrac{-16r + 28}{-36r + 63} $ Multiply the second expression by $\dfrac{9}{9}$ $ \dfrac{9r - 2}{-4r + 7} \times \dfrac{9}{9} = \dfrac{81r - 18}{-36r + 63} $ Therefore $ a = \dfrac{-16r + 28}{-36r + 63} - \dfrac{81r - 18}{-36r + 63} $ Now the expressions have the same denominator we can simply subtract the numerators: $a = \dfrac{-16r + 28 - (81r - 18) }{-36r + 63} $ Distribute the negative sign: $a = \dfrac{-16r + 28 - 81r + 18}{-36r + 63}$ $a = \dfrac{-97r + 46}{-36r + 63}$ Simplify the expression by dividing the numerator and denominator by -1: $a = \dfrac{97r - 46}{36r - 63}$